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Tasks & Questions
- 1. Ingredients for Successful Lessons: Challenging Tasks & Questions that Matter Gail Burrill Michigan State University [email protected]
- 2. Triangles, perimeter & area Dick & Hollebrands, 2011
- 4. Gliders Two different gliders start from the same height. Which glider goes farther: one with a glide ratio of 25/185 or one with a glide ratio of 20/155? Explain your reasoning.
- 6. Characteristics of tasks Multiple representations Multiple strategies for solutions Multiple solutions Multiple entry points Models to develop concepts Cognitive demand - require critical thinking Connections among strands, concepts Different contexts for same concept Progressive formalization
- 7. Characteristics of tasks Multiple representations Multiple strategies for solutions Multiple solutions Multiple entry points Models to develop concepts Cognitive demand - require critical thinking Connections among strands, concepts Different contexts for same concept Progressive formalization
- 8. Opportunities for discussion Tasks have to be justified in terms of the learning aims they serve and can work well only if opportunities for pupils to communicate their evolving understanding are built into the planning. (Black & Wiliam, 1998)
- 9. Mathematical Practices: How students should work Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning CCSS, 2010
- 10. Mathematical Practices: How students should work Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning CCSS, 2010
- 11. More triangles Draw a triangle ABC Construct the perpendicular bisector of side AB Construct the perpendicular bisector of side BC Make a conjecture about the perpendicular bisector of side AC. Move point A What do you observe?
- 12. Characteristics of tasks- which were present in last task as given? Multiple representations Multiple strategies for solutions Multiple solutions Multiple entry points Models to develop concepts Cognitive demand - require critical thinking Connections among strands, concepts Different contexts for same concept Progressive formalization
- 13. A rubric for inquiry math tasks Harper & Edwards, 2011
- 14. The only reasons to ask questions are: (Black et al., 2004) To PROBE or uncover students’ thinking. • understand how students are thinking about the problem. • discover misconceptions. • use students’ understanding to guide instruction. To PUSH or advance students’ thinking. • make connections • notice something significant. • justify or prove their thinking.
- 15. 00:04:17 T …two cubed, …the base is gonna be multiplied three times. 00:04:22 T Two, times two, times two. 00:04:25 T And the thing we're gonna learn about …is exponential growth. 00:04:29 T …we have two cubes. This would be like two to the first power. 00:04:34 T So if we made it two squared, which would be two times two, we would see that it grows to two squared. That's two times two, right? 00:04:44 T Two cubed is two, times two, times two. Two to the third power… 00:04:53 T Then if we go two to the fourth, you're looking at... 00:05:05 T Now two to the fourth is how much? 00:05:08 SN Sixteen. 00:05:09 T Sixteen. 00:05:14 T Okay. So two to the fifth would be how much? 00:05:17 SN Twenty-five. 00:05:18 SN Twenty-five? 00:05:19 SN No. 00:05:20 SN Twenty. 00:05:21 SN Thirty-two. 00:05:24 T Two to the fourth is 16…. 00:05:26 T And we take that and multiply it by two and we get? 00:05:28 SS Thirty-two. 00:05:29 T Thirty-two. Okay. NCES TIMSS US Video 1999
- 16. Making It Happen - Facilitating Discourse What would you predict would happen, you are not actually going to do it, what would you predict would happen? Group 3, you guys think… think that, you guys said that when you slant the sides the area stays the same and so do the sides but…the angles change. Why do you think that is true. Can you show us up there? Try that, I don't know? So, what does that mean? How did you measure the area? So, so what she is saying is that…what's the formula again? ..so you're saying the length times the width….lets soak that in for a second. Bringing it all together video clip, 2012
- 17. Reasoning/Sense Making Questions Compare and contrast Predict forward Predict backward Analyze a connection/relationship Generalize/make conjectures Justify/prove mathematically Consider assumptions inherent in the problem and what would happen if they were changed Interpret information, make/ justify conclusions Burrill & Dick, 2008
- 18. What we do with tasks Setting up Adaptation/modification Implementation Respond to student questions Prompts Monitor student work Discussion Choose solutions to share Sequence solutions to meet mathematical goal Manage solution strategies Ask questions Consolidate the math using student work (Stein & Smith, 2011)
- 19. Equivalent Expressions Which expression is equivalent to 3(8x-2y+7)? • 24x-2y+7 • 24x-6y+21 • 8x-6y+21 • 11x-5y+10 Albert wants to simplify the expression: 8(3–y) + 5(3–y). Which of the following is equivalent to the expression above? A. 39 – y B. 13(3 – y) C. 40(3 – y) D. 13(6 – 2y)
- 20. Equivalent Expressions Which expression is equivalent to 3(8x-2y+7)? 23%* A. 24x-2y+7 33% B. 24x-6y+21 Michigan 2007, Gr. 8 26% C. 8x-6y+21 18% D. 11x-5y+10 Albert wants to simplify the expression: 8(3–y) + 5(3–y) Which of the following is equivalent to the expression above? 29% A. 39 – y 40%* B. 13(3 – y) 7% C. 40(3 – y) (Florida 2006, grade 9) 24% D. 13(6 – 2y)
- 23. Examples Systems of equations What is a solution? Inequalities Fix area and perimeter Fibonacci Area from calculus
- 24. Systems of equations Algebra Nspired, 2009
- 25. Questions Matter Algebra Nspired, 2009
- 27. Find a QuickTimeª and a decompressor are needed to see this picture. shape with QuickTimeª and a decompressor are needed to see this picture. an area of 4 sq units. Justify your reasoning. Adapted from Calculus Nspired, 2010
- 28. Students learn if they are actively involved in choosing and evaluating strategies, considering assumptions, and receiving feedback. encounter contrasting cases- notice new features and identify important ones. struggle with a concept before they are given a lecture develop both conceptual understandings and procedural skills National Research Council, 1999; 2001
- 29. Mathematical Practices: How students should work Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning CCSS, 2010
- 30. References Algebra Nspired (2009). Texas Instruments Education Technology. www.ti-mathnspired.com/ Black, P. & Wiliam, D. (1998). “Inside the Black Box: Raising Standards Through Classroom Assessment”. Phi Delta Kappan. Oct. pp. 139-148. Black, P. Harrison, C., Lee, C., Marshall, E., & Wiliam, D. (2004). “Working Inside the Black Box: Assessment for Learning in the Classroom,” Phi Delta Kappan, 86 (1), 9-21. Bringing it all together. (2012). Video clip shown at the 2012 annual National Council of Supervisors of Mathematics annual meeting, Philadelphia PA. film by Brennan, B., Olson J. & the Janus Group. Curriculum Research & Development Group. University of Hawai’i at Manoa, Honolulu HI (2009). Burrill, G. & Dick, T. (2008). What state assessments tell us about student achievement in algebra. Paper presented at NCTM 2008 Research Presession Calculus Nspired (2009). Texas Instruments Education Technology. www.ti-mathnspired.com/
- 31. Common Core Standards. College and Career Standards for Mathematics 2010). Council of Chief State School Officers (CCSSO) and (National Governor’s Association (NGA) Dick, T., & Hollebrands, K. (2011). Focus in high school mathematics: Technology to support reasoning and sense. Reston VA: National Council of Teachers of Mathematics Florida Department of Education (2006). FCAT Mathematics Released Items, Grade 9. Harper, S., & Edwards, T. (2011). A new recipe: No more cookbook lessons. The Mathematics Teacher. 105(3). Pp 180-188. Looking at an angle. (2003). Mathematics in Context Project. Directed by Tom Romberg & Jan deLange.Vhicago IL: Encyclopedia Britannica. Michigan Department of Education. (2007). Released item mathematics grades 8 fall. www.michigan.gov/mde/0,1607,7-140-22709_31168_31355-95470--,00.ht ml National Center for Education Statistics (NCES). (2003).Third International Mathematics and Science Study (TIMSS), Video Study. U.S. Department of Education. http://nces.ed.gov/timss/video.asp National Research Council. (1999). How People Learn: Brain, mind, experience,and school. Bransford, J. D., Brown, A. L., & Cocking, R. R.
- 32. National Research Council (2001). Adding It Up. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) Washington DC: National Academy Press. Also available on the web at www.nap.edu. Smith, M. S., & Stein, M.K., (2011). The five practices for organizing productive mathematical discussions. Reston, VA: National Council of Teachers of Mathematics.
Editor's Notes
- #10: The mathematical practices should be woven through every lesson as appropriate - these are ways of thinking about and doing mathematics that should permeate all of the work students do- and should not be taught in an isolated - today we will model with mathematics. The examples that follow suggest how these practices might be incorporated using some Nspired activities.
- #11: The mathematical practices should be woven through every lesson as appropriate - these are ways of thinking about and doing mathematics that should permeate all of the work students do- and should not be taught in an isolated - today we will model with mathematics. The examples that follow suggest how these practices might be incorporated using some Nspired activities.
- #20: Can use these to motivate the activity- can leave out if want
- #30: The mathematical practices should be woven through every lesson as appropriate - these are ways of thinking about and doing mathematics that should permeate all of the work students do- and should not be taught in an isolated - today we will model with mathematics. The examples that follow suggest how these practices might be incorporated using some Nspired activities.